One significant subfield of additive group theory and combinatorial number theory is the zero-sum theory that studies the sums behavior of suitable sequences of elements in an abelian finite group \(G\)  (see, for instance, the surveys [1]}, [2]}). In this context, a typical kind of problem considers the existence of constants \(\ell \)  such that any sequence of elements of \(G\)  whose length is bigger than \(\ell \)  satisfies an additive property \(\mathcal {P}\) . Among these constants, an important role is taken by the classical Erdős-Ginzburg-Ziv constant of a group \(G\)  that denotes the smallest positive integer \(\ell \)  such that any sequence of length \(|S|\ge \ell \)  contains a zero-sum subsequence of length \(|G|\) . This constant has been well studied in the literature, we refer to the survey paper [2]}. Here we recall that in [4]}, Erdős, Ginzburg and Ziv completely determined its value over cyclic groups and that in [5]} and in [6]}, respectively Fox, Sauermann and Naslund, derived nontrivial upper bounds on groups of type \(\mathbb {F}_p^n\)  (they assumed a slightly different definition of Erdős-Ginzburg-Ziv constant).
